Abstract

One of the fascinating topological phenomena is the edge state in one-dimensional system. In this work, the topological photonics in the dimer chains composed by the split ring resonators are revealed based on the Su-Schrieffer-Heeger model. The topologically protected photonic edge state is observed directly with the in situ measurements of the local density of states in the topological nontrivial chain. Moreover, we experimentally demonstrate that the edge state localized at both ends is robust against a varied of perturbations, such as losses and disorder. Our results not only provide a versatile platform to study the topological physics in photonics but also may have potential applications in the robust power transfer.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Topological photonics have attracted a lot of interest for the application in the robust one-way transmission. The rise of topological photonics [1, 2], is along with the discovery of various topological phases in condensed matter physics. Recently, photonic topological one-way edge modes with broken time-reversal (T-reversal) symmetry have been widely studied in theory [3] and experiment [4] based on photonic crystals composed by gyro-magnetic materials and helical optical fibers, respectively. The quantum spin Hall effect (QSHE) [5], T-reversal symmetric electron systems with nontrivial topological properties, also attracted people’s great attention for its novel spin-dependent topological phases. Inspiring by the QSHE of electron, up to now, a variety of optical analogues have been proposed, by use of metamaterials [6,7] and two-dimension (2-D) photonic crystals [8–10]. In addition to the edge state in the 2-D system, the topological states in the one dimensional (1-D) system also have been demonstrated in metamaterials [11], resonant structures [12–14], photonic/phononic crystals [15–17], and 1-D waveguide array [18–20]. The topological interface state between two crystals with distinct topological gap has been demonstrated, which can be used for the field enhancement [16]. Very recently, this interface state has been used to determine the topological invariants of the polaritonic quasicrystals [21], and the chiral edge states in 1-D double coupled Peierls chain have been observed for the first time [22].

Topological states are protected by the topological phase transition across the interface. Thus they are robust against the defects, the losses, and the disorder [13, 23]. In 1D system, the robustness of the topological interface state has been investigated in dielectric resonator chains [13]. Su-Schrieffer-Heeger (SSH) array is one of the standard tight-binding models that have attracted people’s great interest in the topological band-gap modes [20, 25–28]. In this work, we experimentally demonstrate the robust edge states of the SSH model in a split-ring-resonator (SRR) chain. SRRs with broken rotational symmetry provide a new degree of freedom, the azimuth in addition to the distance and the background between the resonators, to adjust the coupling in the chain. Using the near-field method, we experimentally get the density of states (DOS) of two dimer chains with different topological property, and demonstrate that the edge states exist in the topological nontrivial chain. We systemically studied the robustness of edge states which are insensitive to a varied of perturbations, such as losses and disorder in the structure. Our results provide a versatile platform to observe the robust topological edge state. In addition, the robust edge states at the two ends of a chain may have some potential applications in information transmission, power transfer, and topological gap soliton.

2. Two different dimer chains realized by SRRs and the edge state in the topological structure

Our experimental setup is shown in the Fig. 1. All the SRRs are identical with the same resonant frequency of ω0=1.9 GHz, which is determined by the geometric parameters, including the thickness w=1.0mm, the height h=5.0mm, the inner radius r=10mm, and the gap sizeg=1.5mm (Details of the experimental setup and measurement can be found in Appendix A). The unit cell of the one-dimensional dimer chains consists of two SRRs whose gaps have different azimuth angles, and the chain is composed of 16 unit cells in experiments. The sample is sandwiched by two metal plates in measurements (the top metal plate is removed to take a picture of the sample). The two metal plates separated by 30 mm act as a waveguide with the cutoff frequency of 5 GHz for polarization parallel to the plate. Below the cutoff frequency, there is only evanescence wave in bare waveguide. Therefore, the bare waveguide can be regarded as an electromagnetic “topological trivial insulator”. Meanwhile, the interaction between SRRs can only rely on the near-field duo to the suppression of the far-field radiation. So our system works well in the tight-binding regime.

 

Fig. 1 Experimental setup. The one dimensional dimer chain composed by equally spaced 32 identical SRRs (not shown all the SRRs) is arranged on a foam substrate, and sandwiched by two metallic plates in experiments (here the top metal plate is taken away in order to take a picture of the chain). The near-field probe made of a non-resonant loop is used to measure the density of states.

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Considering only the nearest-neighbor coupling, the equation of motion for the infinite dimer chain can be described as (Details can be found in Appendix C):

ωnor2(akbk)=(1κintra+κintereikdκintra+κintereikd1)(akbk),
where ωnor denotes the frequency, which is normalized with respect of the resonant frequency ω0, (ak,bk)T represents the cell-periodic Bloch current Eigen-function of a state with wave-vector k, d is the lattice constant, κintra and κinter are the intra-dimer and the inter-dimer coupling strengths, respectively. It is essential to manipulate the coupling between resonators in experimental investigation of edge states in SSH model. A straightforward way is to resort to the dependence of coupling strength on distance [13, 24]. Other methods include alternating the electromagnetic background between resonators [18, 29] and utilizing the azimuthally dependent coupling between the dipoles [12]. In our setup, the rotational symmetry of the resonator is broken due to the gap, so that the coupling strength can be flexibly adjusted by in-plane rotation of the SRRs.

Sketch of a pair of arbitrarily rotated split rings is shown in Fig. 2(a). The rotating angles of the rings are φ1 and φ2, respectively. The calculated relationship between rotating angles and the coupling strengthen are given in Fig. 2(b). Details of the calculation can be found in Appendix B. It is found that when φ1,2 is near 180° the coupling is very sensitive to the angles. In contrast, when φ1,2 is near 0° the coupling is not sensitive to the angles. In addition, the coupling strength can be zero at the proper rotation angles, although the distance between two rings is very short. Obviously, the coupling strength of the case whose gaps in neighboring SRRs are next to each other (φ1=φ2=180o), is stronger than the case whose gaps in neighboring SRRs are on opposite sides (φ1=φ2=0o). When the distance between two SRRs is fixed to d/2 = 24 mm, the strong (weak) coupling parameter is 0.48 (−0.21) [30].

 

Fig. 2 (a) Sketch of a pair of arbitrarily rotated split rings. (b) Controlling coupling strengthen realized by tuning the relevant angle between two resonators. The strong and weak coupling strength used in our SSH chains are shown in the upper inset and down inset respectively.

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We first design two different dimer chains. For convenience, the chains with |κintra|>|κinter| are called typeI, and the ones with |κ'intra|<|κ'inter| are called type II. They can emulate polyethylene ending with strong bond and with weak bond, respectively. We first consider the type I (topological trivial) chain with κintra=0.48, κinter=0.21, whose unit cell is shown in the inset of Fig. 3(a). By using Eq. (1), the calculated Eigen frequencies of the finite chain with 16 unit cells are given with black dots in Fig. 3(a). One can find there are two isolated bands separated by a gap, which is indicated by the gray area. In experiments, the DOS of this Type I sample in the band is relatively high, whereas in the gap it is almost zero, as shown in Fig. 3(a). It is consistent with the theoretical calculation (marked by the black dots). Here the DOS spectrum is obtained by averaging the local density of states (LDOS) spectral over all sites, and the LDOS spectrum of each site is obtained from the reflection by putting the probe to the center of the corresponding SRR [31]. Here all of the LDOS measurements have been normalized.

 

Fig. 3 Two types of dimer chains differ in their topological properties. (a-b) Schematic of the unit cell, calculated Eigen frequencies (black dots), and measured DOS spectrum (blue profile) of the Type I and the Type II chains are given in (a) and (b), respectively. (c) Experimental (red triangles) and theoretical (grey dashed line) LDOS distribution of the edge state (f = 1.9GHz). (d) LDOS profiles of the ordinary state in pass band (f = 1.95GHz).

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Next, we study a 16-unit type II chain (topological non-trivial) with κ'intra=0.21,κ'inter=0.48, whose unit cell is shown in the inset of Fig. 3(b). Similarly, we calculate its Eigen frequencies and measure its DOS spectrum, which consist with each other well. Compared with the results of the type I chain, there is an additional state in the gap for the type II chain. Calculations and measurements show that the LDOS of the new state is strongly localized at the two ends, as shown in Fig. 3(c). Hence it belongs to the edge states. It is totally different from the ordinary state in the band whose LDOS is mainly distributed in the bulk [see in Fig. 3(d)].

The topological property of 1-D system can be characterized by the winding number of the band [32, 33] (Details can be found in Appendix D):

w=12ππ/dπ/dθkdk,
where θk is the polarization vector angle. After calculation, we get the winding number for both upper and lower bands are w=0 for the typeIchain, and w'=1 for the typeII infinite long chain. Zak phase of the bands can be directly obtained as ϕZak=πw [33]. As the corresponding relationship between the band gap and the passband, the gaps of two chains considered above are completely different [16]. The band gap of type I chain is trivial as the bare waveguide, while the band gap of type II chain is nontrivial. The edge states observed at the two ends of the type II chain are topologically protected by the topological transition-from trivial bare waveguide to nontrivial dimer chain. The robustness of the edge states is investigated experimentally in the following section.

3. Experimental demonstration of the robust edge states against losses and disorder

In this section, we will reveal that the edge state in the typeII chain is robust against certain losses and disorder. At first, we add the losses into the central 20 SRRs of the chain as shown in Fig. 4(a). The lossy SRRs marked with the grey background are realized by adding absorbing material into the interior of the rings. Measured DOS spectrum is shown in Fig. 4(b). One can clearly see that the ordinary bulk states are affected significantly while the edge state in the gap region (grey area) is almost immune to the loss perturbations. In order to further illustrate the unique of the robust edge state, the LDOS distributions in the lossless and lossy chains are compared in Figs. 4(c) and 4(d). The measured LDOS of the edge state is still confined at the two ends [the red dots in Fig. 4(c)], just as the lossless system [black solid line in Fig. 4(c)]. While for the bulk state, the LDOS is significantly affected, as shown in Fig. 4(d).

 

Fig. 4 Robust edge state against the losses perturbation. (a) The topological nontrivial chain with losses added into the central 20 SRRs (indicated by the grey background). (b) Measured DOS spectrum with loss. DOS of the edge state is much more robust than that of the bulk state. (c) Measured LDOS distribution of the edge state (f = 1.9GHz) with losses (red triangles), along with the theoretical calculations without losses (grey dashed line). (d) Measured LDOS distribution of the ordinary bulk state (f = 1.95GHz) with losses (red triangles), along with the theoretical calculations without losses (grey dashed line).

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Secondly, we investigate the robustness of the edge states against certain disorder perturbation. The structure disorder is realized by rotating the central 20 SRRs, as shown in Fig. 5(a). The detail of rotation is illustrated in the inset of Fig. 5(b). Three disorder chains are considered, in which the central 20 SRRs are random rotated of α=1, 3 and 5 degrees, respectively. Measured LDOS distributions of the edge state and the bulk state in three chains are shown in Figs. 5(b) and 5(c), respectively. By comparing with results of the original chain (black solid line), one can find that at different disorder level the edge state is still maintained, while the bulk state has been deteriorated seriously.

 

Fig. 5 Robust edge states against the disorder perturbation. (a) Schematic representation of the topological nontrivial chain with disorder (random coupling strengths by rotating the central 20 SRRs). Detail of rotation is shown in the inset of Fig. 4(b). (b, c) Measured LDOS distributions of the edge state and the bulk state at various disorder levels, including α=1° (red circles), α=3° (blue stars), and α=5° (green triangles). As a comparison, the calculated LDOS distribution of the edge (f = 1.9GHz) and the bulk (f = 1.95GHz) states in the original chain are also presented (gray dashed line).

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At last, the robustness of the edge state is further examined by adding the losses and the structure disorder simultaneously. By comparing three structures with both losses and random rotate angles in the central 20 SRRs (just as the situation in Fig. 5(a)), we find the topological edge state is still maintained, as shown in Fig. 6. Remarkably, this demonstration of topological robust effect does not resort to the structural disorder by randomly distributing the inter-site separations [13] and may open novel routes to exploit new ways for steering the random distribution.

 

Fig. 6 Measured LDOS distributions of the edge states in the chain with both losses and disorder perturbations in the central 20 SRRs. As a comparison, the calculated LDOS distribution of the edge state (f = 1.9GHz) in the original chain are also presented (grey dashed line).

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In discussion, our results have revealed that the topological edge states cannot be affected when certain losses and random rotation are introduced to the center 20 SRRs of the structure instead of the specific positions. This may have some significant applications, such as the robust wireless transfer of information and power. Common wireless power transfer (WPT) scheme involve transmitter and receiver coils that are magneto-inductively coupled together [38]. Here by applying a source to the left end of the topological nontrivial SSH chain, the topological edge state will be established with near-field location around the two ends. One can image that if there is a non-resonant loop with a load at the right end, the harmonic magnetic field localized at the right end will generates the induced electromotive force in that loop. That is the power transfer from the left end to the right end using edge state. In such near-field system, the power transfer and the communication are based on the near-filed coupling instead of the far-filed propagating. Specially, this power transfer manner is immune to the structural disturbances such as the disorders and losses in the middle part of the chain.

4. Conclusion

In summary, we experimentally demonstrate the edge states of the SSH model by in situ measurements of the local density of states in a dimer SRR chain, in which SRRs with broken rotational symmetry provide the new azimuth degree of freedom to adjust the coupling between the resonators. It is observed that the edge states are robust against a variety of perturbations, such as the losses and the disorder. Our results not only provide a versatile platform to study the robust topological edge state, but also may contribute potential application in the information transmission, power transfer, and so on. Although our results are from 1-D system, the array of split ring resonators can even be used to explore the topological phenomena in 2-D system.

Appendix

A. The details of experimental methods

Our resonators and the waveguide are fabricated in the precision workshop. The materials of the resonators and the plates of waveguide are copper and aluminum, respectively. In the experiment, we firstly accurately arrange the resonators in the waveguide according to the printed drawing corresponding to the designed patterns. Specially, a layer of foam plate with the thickness of 10 mm, which has the same dielectric constant as air in microwave regime, is placed on the bottom plate of the waveguide as the substrate. Our near-field probe is a home-made loop antenna, which is connected to the port of the vector network analyzer (Agilent PNA Network Analyzer N5222A). The radius of the loop probe is 2 mm. It can be taken as a non-resonant antenna with high impedance. We use this small loop antenna as a source to excite our samples, and then measure the reflection. In order to accurately probe the reflection spectrum associated with the LDOS, our experimental samples are placed on an automatic translation device. The spatial precision of scanning is 0.1 mm.

B. The coupling strength between two SRRs

The coupling between two SRRs is composed of magnetic and electrical coupling [30]. So we should calculate two types of coupling separately and then obtain the total coupling strength of two SRRs. Here we determine the coupling coefficient with the aid of the charge and current densities in the split rings. The mutual inductance M can be get from the equation [30]:

M=μ04πIm1(r1)Im2(r2)|r1r2|ds1ds2,
where μ0 is the permeability in free space. Iml(l=1,2) is a dimensionless quantity and takes the maximum one when the angular coordinate is θ=0o (see Fig. 7(a)). rl is its position vector. ds1 and ds2 are respectively the line element along the filament of the left and right rings (see Fig. 7(b)). Similarly, the mutual capacitance is obtained as [34, 35]
K1=14πε0ρm1(r1)ρm2(r2)|r1r2|dτ1dτ2,
where ε0 is the free-space permittivity. dτi(i=1,2) denotes the volume element of two SRRs. ρml(l=1,2) means the corresponding normalized line charges. For evaluating the integrals, we need functional relationships for the filament currents and for the charge distribution. For finding the coupling coefficients, we need the self-capacitance and self-inductance of the rings and also the expression for the charge. The charge distribution in the split ring is given as [36]
ρm(θ)=(tanθ2)/2ln(g4r), (θ<θg
where ±θg are the coordinates of the gap (see Fig. 7(a)). And the normalized current is of the form [36]

 

Fig. 7 (a) The schematic of a single SRR. (b) A pair of coupled SRRs in the planar configuration. The angle of rotation of two resonators are marked by φ1 and φ2.

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Im(θ)=ln(cosθg2cosθ2)ln1(cosθg2).

The approximate values of the self-inductance and self-capacitance are [37]

L=μ0r(ln8rh+w12),
and

C=ε0[(w+g)(h+g)g+2(h+w)πln(4rg)].

Up to now, we have got all the information to determine the magnetic and electric contributions to the total coupling coefficient with

κH=2ML,κE=2CK.

The total coupling is κ=κE+κH .

C. The equation of motion for the dimer chain system

We next deduce the equation of motion for the SSH chain shown in supplementary Fig. 1(c). With tight-binding approximation, the Lagrangian of the chain may be written as follows:

l=n[L2(p˙n2+q˙n2)pn2+qn22C+M2p˙nq˙n1pnqn1K2pnqnK1+M1p˙nq˙n]. (A. 8)
where pn and qn are the charges on the rings in the nth unit cell, L (C) and Ml (Kl) are the self-inductance (capacitance) and mutual inductance (capacitance). Without considering the Ohm resistance, the Euler equation can be written as follows:

ddt(lα˙n)lαn=0,(α=p,q).

Assuming propagating solutions for both sub-lattice, the currents on two rings can be expressed as: p˙n=kBZakexp[i(kndωt)],q˙n=kBZbkexp[i(kndωt)], respectively. We arrive at equation of motion for currents:

ddt(Lp˙n+M2q˙n1+M1q˙n)+1iωCp˙n+1iωK2q˙n1+1iωK1q˙n=0ddt(Lq˙n+M2p˙n+1+M1p˙n)+1iωCq˙n+1iωK2p˙n+1+1iωK1p˙n=0.

Written in a more compact form:

ωnor2[κH22+κH12+2cos(kd)κH2κH11](akbk)=H'(akbk).
with H'=(κH2κE2κH2κE1eikdκE2κH1eikdκH1κE11κ1+κ2eikdκ1+κ2eikdκH2κE2κH2κE1eikdκE2κH1eikdκH1κE11),where ωnor=ω/ω0 is the frequency normalized to the resonant frequency ω0=1/LC,κEl=C/Kl (κHl=2Ml/L) is the electric (magnetic) coupling coefficient, κ1=κH1+κE1,κ2=κH2+κE2.

In our experiment, we start from the case that φ1=φ2=0o, and then rotating all split rings anticlockwise simultaneously to change the chain to a different topological phase. In this process, we calculate all the values of κs1=κH22+κH12+2κH2κH1,κs2=κH22+κH122κH2κH1,κs3=κH2κE2κH1κE1,κs4=κH2κE1 and κs5=κE2κH1 in terms of rotation radian as shown in Fig. 8. From the results, κsl are negligible relative to 1 and the Eq. (A.11) can be written in a simplified form as follows:

ωnor2(akbk)=(1κ1+κ2eikdκ1+κ2eikd1)(akbk),
where κ1 and κ2 denote the total coupling within a dimer and between two dimers.

 

Fig. 8 The relationship between κsl (l = 1, 2, 3, 4, and 5) and the rotation angle of the pair of SRRs.

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D. The calculation method of winding number

For our system, the equation of motion in wave-vector space is governed by

ωnor2(akbk)=Dk(akbk),
with Dk=(1κ1κ2eikdκ1κ2eikd1). The dynamical matrix Dk play the same role of the Hamiltonian. The diagonal element is the onsite potential. κ1 (κ2) is dimensionless and denotes the intra-dimer (inter-dimer) coupling strength [22].

The eigenvectors of Eq. (A.13) for both upper and lower band are expressed as

|uk,±=(akbk)=(κ1+κ2cos(kd)iκ2sin(kd)(κ1+κ2cos(kd))2+(κ2sin(kd))21),

Here we rewrite Eq. (A.14) as follows:

|uk,±=(eiθk1),

Where θk=arctan(κ2sin(kd)κ1+κ2cos(kd)), and its normalized form is

|uk,±=12(eiθk1),
The winding number is the number of loops made by a Bloch state around the equator of the Bloch sphere, as k passes through the Brillouin zone. For our model, it can be expressed as follows:

wwinding=iππ/dπ/d(ak*kak+bk*kbk)dk.

Combine Eq. (A.16) and Eq. (A.17), we get

w=iππ/dπ/d(ak*kak+bk*kbk)dk=12ππ/dπ/dθkdk,

By numerical calculation, the winding numbers for both upper and lower band are 1 when |κ1|<|κ2|, while they are 0 when |κ1|>|κ2|.

Funding

National Key Research Program of China (2016YFA0301101); National Natural Science Foundation of China (NSFC) (11674247, 61621001, 11504236, 11474220); Natural Science Foundation of Shanghai (18ZR1442900, 17ZR1443800); and the Fundamental Research Funds for the Central Universities.

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33. S. Q. Shen, Topological Insulators: Dirac Equation in Condensed Matters (Springer Science & Business Media, 2012).

34. L. D. Landau and E. M. Lifschitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1984).

35. K. Simonyi, Foundations of Electrical Engineering (Pergamon, Oxford, 1963).

36. J. E. Allen and S. E. Segre, “The electric field in single-turn and multi-sector coils,” Nuovo Cim. 21(6), 980–987 (1961). [CrossRef]  

37. O. Sydoruk, E. Tatartschuk, E. Shamonina, and L. Solymar, “Analytical formulation for the resonant frequency of split rings,” J. Appl. Phys. 105(1), 014903 (2009). [CrossRef]  

38. A. Kurs, A. Karalis, R. Moffatt, J. D. Joannopoulos, P. Fisher, and M. Soljačić, “Wireless power transfer via strongly coupled magnetic resonances,” Science 317(5834), 83–86 (2007). [CrossRef]   [PubMed]  

References

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  1. L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8(11), 821–829 (2014).
    [Crossref]
  2. L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological states in photonic systems,” Nat. Phys. 12(7), 626–629 (2016).
    [Crossref]
  3. F. D. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100(1), 013904 (2008).
    [Crossref] [PubMed]
  4. Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljacić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461(7265), 772–775 (2009).
    [Crossref] [PubMed]
  5. C. L. Kane and E. J. Mele, “Quantum spin Hall effect in graphene,” Phys. Rev. Lett. 95(22), 226801 (2005).
    [Crossref] [PubMed]
  6. A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2013).
    [Crossref] [PubMed]
  7. Z. Guo, H. Jiang, Y. Long, K. Yu, J. Ren, C. Xue, and H. Chen, “Photonic spin Hall efect in waveguides composed of two types of single-negative metamaterials,” Sci. Rep. 7(1), 7742 (2017).
    [Crossref] [PubMed]
  8. M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photonics 7(12), 1001–1005 (2013).
    [Crossref]
  9. G. Q. Liang and Y. D. Chong, “Optical Resonator Analog of a two-dimensional topological insulator,” Phys. Rev. Lett. 110(20), 203904 (2013).
    [Crossref] [PubMed]
  10. L. H. Wu and X. Hu, “Scheme for achieving a topological photonic crystal by using dielectric material,” Phys. Rev. Lett. 114(22), 223901 (2015).
    [Crossref] [PubMed]
  11. W. Tan, Y. Sun, H. Chen, and S. Q. Shen, “Photonic simulation of topological excitations in metamaterials,” Sci. Rep. 4(1), 3842 (2015).
    [Crossref] [PubMed]
  12. A. P. Slobozhanyuk, A. N. Poddubny, A. E. Miroshnichenko, P. A. Belov, and Y. S. Kivshar, “Subwavelength topological edge States in optically resonant dielectric structures,” Phys. Rev. Lett. 114(12), 123901 (2015).
    [Crossref] [PubMed]
  13. C. Poli, M. Bellec, U. Kuhl, F. Mortessagne, and H. Schomerus, “Selective enhancement of topologically induced interface states in a dielectric resonator chain,” Nat. Commun. 6(1), 6710 (2015).
    [Crossref] [PubMed]
  14. C. W. Ling, M. Xiao, C. T. Chan, S. F. Yu, and K. H. Fung, “Topological edge plasmon modes between diatomic chains of plasmonic nanoparticles,” Opt. Express 23(3), 2021–2031 (2015).
    [Crossref] [PubMed]
  15. M. Xiao, Z. Q. Zhang, and C. T. Chan, “Surface impedance and bulk band geometric phases in one-dimensional systems,” Phys. Rev. X 4(2), 021017 (2014).
    [Crossref]
  16. M. Xiao, G. Ma, Z. Yang, P. Sheng, Z. Q. Zhang, and C. T. Chan, “Geometric phase and band inversion in periodic acoustic systems,” Nat. Phys. 11(3), 240–244 (2015).
    [Crossref]
  17. W. S. Gao, M. Xiao, C. T. Chan, and W. Y. Tam, “Determination of Zak phase by reflection phase in 1D photonic crystals,” Opt. Lett. 40(22), 5259–5262 (2015).
    [Crossref] [PubMed]
  18. N. Malkova, I. Hromada, X. Wang, G. Bryant, and Z. Chen, “Transition between Tamm-like and Shockley-like surface states in optically induced photonic superlattices,” Phys. Rev. A 80(4), 043806 (2009).
    [Crossref]
  19. A. Blanco-Redondo, I. Andonegui, M. J. Collins, G. Harari, Y. Lumer, M. C. Rechtsman, B. J. Eggleton, and M. Segev, “Topological optical waveguiding in silicon and the transition between topological and trivial defect states,” Phys. Rev. Lett. 116(16), 163901 (2016).
    [Crossref] [PubMed]
  20. S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, “Topologically protected bound states in photonic parity-time-symmetric crystals,” Nat. Mater. 16(4), 433–438 (2017).
    [Crossref] [PubMed]
  21. F. Baboux, E. Levy, A. Lemaître, C. Gómez, E. Galopin, L. L. Gratiet, I. Sagnes, A. Amo, J. Bloch, and E. Akkermans, “Measuring topological invariants from generalized edge states in polaritonic quasicrystals,” Phys. Rev. B 95(16), 161114 (2017).
    [Crossref]
  22. S. Cheon, T. H. Kim, S. H. Lee, and H. W. Yeom, “Chiral solitons in a coupled double Peierls chain,” Science 350(6257), 182–185 (2015).
    [Crossref] [PubMed]
  23. C. Liu, W. Gao, B. Yang, and S. Zhang, “Disorder-induced topological state transition in photonic metamaterials,” Phys. Rev. Lett. 119(18), 183901 (2017).
    [Crossref] [PubMed]
  24. S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101(4), 047401 (2008).
    [Crossref] [PubMed]
  25. L. H. Li, Z. H. Xu, and S. Chen, “Topological phases of generalized Su-Schrieffer-Heeger models,” Phys. Rev. B 89(8), 085111 (2014).
    [Crossref]
  26. E. J. Meier, F. A. An, and B. Gadway, “Observation of the topological soliton state in the Su-Schrieffer-Heeger model,” Nat. Commun. 7, 13986 (2016).
    [Crossref] [PubMed]
  27. Y. Hadad, A. B. Khanikaev, and A. Alu, “Self-induced topological transitions and edge states supported by nonlinear staggered potentials,” Phys. Rev. B 93(15), 155112 (2016).
    [Crossref]
  28. Y. Hadad, J. C. Soric, A. B. Khanikaev, and A. Alu, “Self-induced topological protection in nonlinear circuit arrays,” Nat. Electron. 1(3), 178–182 (2018).
    [Crossref]
  29. Z. Guo, H. Jiang, Y. Li, H. Chen, and G. S. Agarwal, “Enhancement of electromagnetically induced transparency in metamaterials using long range coupling mediated by a hyperbolic material,” Opt. Express 26(2), 627–641 (2018).
    [Crossref] [PubMed]
  30. E. Tatarschuk, N. Gneiding, F. Hesmer, A. Radkovskaya, and E. Shamonina, “Mapping inter-element coupling in metamaterials: Scaling down to infrared,” J. Appl. Phys. 111(9), 094904 (2012).
    [Crossref]
  31. M. Bellec, U. Kuhl, G. Montambaux, and F. Mortessagne, “Tight-binding couplings in microwave artificial graphene,” Phys. Rev. B 88(11), 115437 (2013).
    [Crossref]
  32. R. D. King-Smith and D. Vanderbilt, “Theory of polarization of crystalline solids,” Phys. Rev. B Condens. Matter 47(3), 1651–1654 (1993).
    [Crossref] [PubMed]
  33. S. Q. Shen, Topological Insulators: Dirac Equation in Condensed Matters (Springer Science & Business Media, 2012).
  34. L. D. Landau and E. M. Lifschitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1984).
  35. K. Simonyi, Foundations of Electrical Engineering (Pergamon, Oxford, 1963).
  36. J. E. Allen and S. E. Segre, “The electric field in single-turn and multi-sector coils,” Nuovo Cim. 21(6), 980–987 (1961).
    [Crossref]
  37. O. Sydoruk, E. Tatartschuk, E. Shamonina, and L. Solymar, “Analytical formulation for the resonant frequency of split rings,” J. Appl. Phys. 105(1), 014903 (2009).
    [Crossref]
  38. A. Kurs, A. Karalis, R. Moffatt, J. D. Joannopoulos, P. Fisher, and M. Soljačić, “Wireless power transfer via strongly coupled magnetic resonances,” Science 317(5834), 83–86 (2007).
    [Crossref] [PubMed]

2018 (2)

2017 (4)

C. Liu, W. Gao, B. Yang, and S. Zhang, “Disorder-induced topological state transition in photonic metamaterials,” Phys. Rev. Lett. 119(18), 183901 (2017).
[Crossref] [PubMed]

Z. Guo, H. Jiang, Y. Long, K. Yu, J. Ren, C. Xue, and H. Chen, “Photonic spin Hall efect in waveguides composed of two types of single-negative metamaterials,” Sci. Rep. 7(1), 7742 (2017).
[Crossref] [PubMed]

S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, “Topologically protected bound states in photonic parity-time-symmetric crystals,” Nat. Mater. 16(4), 433–438 (2017).
[Crossref] [PubMed]

F. Baboux, E. Levy, A. Lemaître, C. Gómez, E. Galopin, L. L. Gratiet, I. Sagnes, A. Amo, J. Bloch, and E. Akkermans, “Measuring topological invariants from generalized edge states in polaritonic quasicrystals,” Phys. Rev. B 95(16), 161114 (2017).
[Crossref]

2016 (4)

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological states in photonic systems,” Nat. Phys. 12(7), 626–629 (2016).
[Crossref]

A. Blanco-Redondo, I. Andonegui, M. J. Collins, G. Harari, Y. Lumer, M. C. Rechtsman, B. J. Eggleton, and M. Segev, “Topological optical waveguiding in silicon and the transition between topological and trivial defect states,” Phys. Rev. Lett. 116(16), 163901 (2016).
[Crossref] [PubMed]

E. J. Meier, F. A. An, and B. Gadway, “Observation of the topological soliton state in the Su-Schrieffer-Heeger model,” Nat. Commun. 7, 13986 (2016).
[Crossref] [PubMed]

Y. Hadad, A. B. Khanikaev, and A. Alu, “Self-induced topological transitions and edge states supported by nonlinear staggered potentials,” Phys. Rev. B 93(15), 155112 (2016).
[Crossref]

2015 (8)

S. Cheon, T. H. Kim, S. H. Lee, and H. W. Yeom, “Chiral solitons in a coupled double Peierls chain,” Science 350(6257), 182–185 (2015).
[Crossref] [PubMed]

M. Xiao, G. Ma, Z. Yang, P. Sheng, Z. Q. Zhang, and C. T. Chan, “Geometric phase and band inversion in periodic acoustic systems,” Nat. Phys. 11(3), 240–244 (2015).
[Crossref]

W. S. Gao, M. Xiao, C. T. Chan, and W. Y. Tam, “Determination of Zak phase by reflection phase in 1D photonic crystals,” Opt. Lett. 40(22), 5259–5262 (2015).
[Crossref] [PubMed]

L. H. Wu and X. Hu, “Scheme for achieving a topological photonic crystal by using dielectric material,” Phys. Rev. Lett. 114(22), 223901 (2015).
[Crossref] [PubMed]

W. Tan, Y. Sun, H. Chen, and S. Q. Shen, “Photonic simulation of topological excitations in metamaterials,” Sci. Rep. 4(1), 3842 (2015).
[Crossref] [PubMed]

A. P. Slobozhanyuk, A. N. Poddubny, A. E. Miroshnichenko, P. A. Belov, and Y. S. Kivshar, “Subwavelength topological edge States in optically resonant dielectric structures,” Phys. Rev. Lett. 114(12), 123901 (2015).
[Crossref] [PubMed]

C. Poli, M. Bellec, U. Kuhl, F. Mortessagne, and H. Schomerus, “Selective enhancement of topologically induced interface states in a dielectric resonator chain,” Nat. Commun. 6(1), 6710 (2015).
[Crossref] [PubMed]

C. W. Ling, M. Xiao, C. T. Chan, S. F. Yu, and K. H. Fung, “Topological edge plasmon modes between diatomic chains of plasmonic nanoparticles,” Opt. Express 23(3), 2021–2031 (2015).
[Crossref] [PubMed]

2014 (3)

M. Xiao, Z. Q. Zhang, and C. T. Chan, “Surface impedance and bulk band geometric phases in one-dimensional systems,” Phys. Rev. X 4(2), 021017 (2014).
[Crossref]

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8(11), 821–829 (2014).
[Crossref]

L. H. Li, Z. H. Xu, and S. Chen, “Topological phases of generalized Su-Schrieffer-Heeger models,” Phys. Rev. B 89(8), 085111 (2014).
[Crossref]

2013 (4)

M. Bellec, U. Kuhl, G. Montambaux, and F. Mortessagne, “Tight-binding couplings in microwave artificial graphene,” Phys. Rev. B 88(11), 115437 (2013).
[Crossref]

A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2013).
[Crossref] [PubMed]

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photonics 7(12), 1001–1005 (2013).
[Crossref]

G. Q. Liang and Y. D. Chong, “Optical Resonator Analog of a two-dimensional topological insulator,” Phys. Rev. Lett. 110(20), 203904 (2013).
[Crossref] [PubMed]

2012 (1)

E. Tatarschuk, N. Gneiding, F. Hesmer, A. Radkovskaya, and E. Shamonina, “Mapping inter-element coupling in metamaterials: Scaling down to infrared,” J. Appl. Phys. 111(9), 094904 (2012).
[Crossref]

2009 (3)

O. Sydoruk, E. Tatartschuk, E. Shamonina, and L. Solymar, “Analytical formulation for the resonant frequency of split rings,” J. Appl. Phys. 105(1), 014903 (2009).
[Crossref]

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljacić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461(7265), 772–775 (2009).
[Crossref] [PubMed]

N. Malkova, I. Hromada, X. Wang, G. Bryant, and Z. Chen, “Transition between Tamm-like and Shockley-like surface states in optically induced photonic superlattices,” Phys. Rev. A 80(4), 043806 (2009).
[Crossref]

2008 (2)

F. D. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100(1), 013904 (2008).
[Crossref] [PubMed]

S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101(4), 047401 (2008).
[Crossref] [PubMed]

2007 (1)

A. Kurs, A. Karalis, R. Moffatt, J. D. Joannopoulos, P. Fisher, and M. Soljačić, “Wireless power transfer via strongly coupled magnetic resonances,” Science 317(5834), 83–86 (2007).
[Crossref] [PubMed]

2005 (1)

C. L. Kane and E. J. Mele, “Quantum spin Hall effect in graphene,” Phys. Rev. Lett. 95(22), 226801 (2005).
[Crossref] [PubMed]

1993 (1)

R. D. King-Smith and D. Vanderbilt, “Theory of polarization of crystalline solids,” Phys. Rev. B Condens. Matter 47(3), 1651–1654 (1993).
[Crossref] [PubMed]

1961 (1)

J. E. Allen and S. E. Segre, “The electric field in single-turn and multi-sector coils,” Nuovo Cim. 21(6), 980–987 (1961).
[Crossref]

Agarwal, G. S.

Akkermans, E.

F. Baboux, E. Levy, A. Lemaître, C. Gómez, E. Galopin, L. L. Gratiet, I. Sagnes, A. Amo, J. Bloch, and E. Akkermans, “Measuring topological invariants from generalized edge states in polaritonic quasicrystals,” Phys. Rev. B 95(16), 161114 (2017).
[Crossref]

Allen, J. E.

J. E. Allen and S. E. Segre, “The electric field in single-turn and multi-sector coils,” Nuovo Cim. 21(6), 980–987 (1961).
[Crossref]

Alu, A.

Y. Hadad, J. C. Soric, A. B. Khanikaev, and A. Alu, “Self-induced topological protection in nonlinear circuit arrays,” Nat. Electron. 1(3), 178–182 (2018).
[Crossref]

Y. Hadad, A. B. Khanikaev, and A. Alu, “Self-induced topological transitions and edge states supported by nonlinear staggered potentials,” Phys. Rev. B 93(15), 155112 (2016).
[Crossref]

Amo, A.

F. Baboux, E. Levy, A. Lemaître, C. Gómez, E. Galopin, L. L. Gratiet, I. Sagnes, A. Amo, J. Bloch, and E. Akkermans, “Measuring topological invariants from generalized edge states in polaritonic quasicrystals,” Phys. Rev. B 95(16), 161114 (2017).
[Crossref]

An, F. A.

E. J. Meier, F. A. An, and B. Gadway, “Observation of the topological soliton state in the Su-Schrieffer-Heeger model,” Nat. Commun. 7, 13986 (2016).
[Crossref] [PubMed]

Andonegui, I.

A. Blanco-Redondo, I. Andonegui, M. J. Collins, G. Harari, Y. Lumer, M. C. Rechtsman, B. J. Eggleton, and M. Segev, “Topological optical waveguiding in silicon and the transition between topological and trivial defect states,” Phys. Rev. Lett. 116(16), 163901 (2016).
[Crossref] [PubMed]

Baboux, F.

F. Baboux, E. Levy, A. Lemaître, C. Gómez, E. Galopin, L. L. Gratiet, I. Sagnes, A. Amo, J. Bloch, and E. Akkermans, “Measuring topological invariants from generalized edge states in polaritonic quasicrystals,” Phys. Rev. B 95(16), 161114 (2017).
[Crossref]

Bellec, M.

C. Poli, M. Bellec, U. Kuhl, F. Mortessagne, and H. Schomerus, “Selective enhancement of topologically induced interface states in a dielectric resonator chain,” Nat. Commun. 6(1), 6710 (2015).
[Crossref] [PubMed]

M. Bellec, U. Kuhl, G. Montambaux, and F. Mortessagne, “Tight-binding couplings in microwave artificial graphene,” Phys. Rev. B 88(11), 115437 (2013).
[Crossref]

Belov, P. A.

A. P. Slobozhanyuk, A. N. Poddubny, A. E. Miroshnichenko, P. A. Belov, and Y. S. Kivshar, “Subwavelength topological edge States in optically resonant dielectric structures,” Phys. Rev. Lett. 114(12), 123901 (2015).
[Crossref] [PubMed]

Blanco-Redondo, A.

A. Blanco-Redondo, I. Andonegui, M. J. Collins, G. Harari, Y. Lumer, M. C. Rechtsman, B. J. Eggleton, and M. Segev, “Topological optical waveguiding in silicon and the transition between topological and trivial defect states,” Phys. Rev. Lett. 116(16), 163901 (2016).
[Crossref] [PubMed]

Bloch, J.

F. Baboux, E. Levy, A. Lemaître, C. Gómez, E. Galopin, L. L. Gratiet, I. Sagnes, A. Amo, J. Bloch, and E. Akkermans, “Measuring topological invariants from generalized edge states in polaritonic quasicrystals,” Phys. Rev. B 95(16), 161114 (2017).
[Crossref]

Bryant, G.

N. Malkova, I. Hromada, X. Wang, G. Bryant, and Z. Chen, “Transition between Tamm-like and Shockley-like surface states in optically induced photonic superlattices,” Phys. Rev. A 80(4), 043806 (2009).
[Crossref]

Chan, C. T.

W. S. Gao, M. Xiao, C. T. Chan, and W. Y. Tam, “Determination of Zak phase by reflection phase in 1D photonic crystals,” Opt. Lett. 40(22), 5259–5262 (2015).
[Crossref] [PubMed]

M. Xiao, G. Ma, Z. Yang, P. Sheng, Z. Q. Zhang, and C. T. Chan, “Geometric phase and band inversion in periodic acoustic systems,” Nat. Phys. 11(3), 240–244 (2015).
[Crossref]

C. W. Ling, M. Xiao, C. T. Chan, S. F. Yu, and K. H. Fung, “Topological edge plasmon modes between diatomic chains of plasmonic nanoparticles,” Opt. Express 23(3), 2021–2031 (2015).
[Crossref] [PubMed]

M. Xiao, Z. Q. Zhang, and C. T. Chan, “Surface impedance and bulk band geometric phases in one-dimensional systems,” Phys. Rev. X 4(2), 021017 (2014).
[Crossref]

Chen, H.

Z. Guo, H. Jiang, Y. Li, H. Chen, and G. S. Agarwal, “Enhancement of electromagnetically induced transparency in metamaterials using long range coupling mediated by a hyperbolic material,” Opt. Express 26(2), 627–641 (2018).
[Crossref] [PubMed]

Z. Guo, H. Jiang, Y. Long, K. Yu, J. Ren, C. Xue, and H. Chen, “Photonic spin Hall efect in waveguides composed of two types of single-negative metamaterials,” Sci. Rep. 7(1), 7742 (2017).
[Crossref] [PubMed]

W. Tan, Y. Sun, H. Chen, and S. Q. Shen, “Photonic simulation of topological excitations in metamaterials,” Sci. Rep. 4(1), 3842 (2015).
[Crossref] [PubMed]

Chen, S.

L. H. Li, Z. H. Xu, and S. Chen, “Topological phases of generalized Su-Schrieffer-Heeger models,” Phys. Rev. B 89(8), 085111 (2014).
[Crossref]

Chen, Z.

N. Malkova, I. Hromada, X. Wang, G. Bryant, and Z. Chen, “Transition between Tamm-like and Shockley-like surface states in optically induced photonic superlattices,” Phys. Rev. A 80(4), 043806 (2009).
[Crossref]

Cheon, S.

S. Cheon, T. H. Kim, S. H. Lee, and H. W. Yeom, “Chiral solitons in a coupled double Peierls chain,” Science 350(6257), 182–185 (2015).
[Crossref] [PubMed]

Chong, Y.

Z. Wang, Y. Chong, J. D. Joannopoulos, and M. Soljacić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461(7265), 772–775 (2009).
[Crossref] [PubMed]

Chong, Y. D.

G. Q. Liang and Y. D. Chong, “Optical Resonator Analog of a two-dimensional topological insulator,” Phys. Rev. Lett. 110(20), 203904 (2013).
[Crossref] [PubMed]

Collins, M. J.

A. Blanco-Redondo, I. Andonegui, M. J. Collins, G. Harari, Y. Lumer, M. C. Rechtsman, B. J. Eggleton, and M. Segev, “Topological optical waveguiding in silicon and the transition between topological and trivial defect states,” Phys. Rev. Lett. 116(16), 163901 (2016).
[Crossref] [PubMed]

Eggleton, B. J.

A. Blanco-Redondo, I. Andonegui, M. J. Collins, G. Harari, Y. Lumer, M. C. Rechtsman, B. J. Eggleton, and M. Segev, “Topological optical waveguiding in silicon and the transition between topological and trivial defect states,” Phys. Rev. Lett. 116(16), 163901 (2016).
[Crossref] [PubMed]

Fan, J.

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photonics 7(12), 1001–1005 (2013).
[Crossref]

Fisher, P.

A. Kurs, A. Karalis, R. Moffatt, J. D. Joannopoulos, P. Fisher, and M. Soljačić, “Wireless power transfer via strongly coupled magnetic resonances,” Science 317(5834), 83–86 (2007).
[Crossref] [PubMed]

Fung, K. H.

Gadway, B.

E. J. Meier, F. A. An, and B. Gadway, “Observation of the topological soliton state in the Su-Schrieffer-Heeger model,” Nat. Commun. 7, 13986 (2016).
[Crossref] [PubMed]

Galopin, E.

F. Baboux, E. Levy, A. Lemaître, C. Gómez, E. Galopin, L. L. Gratiet, I. Sagnes, A. Amo, J. Bloch, and E. Akkermans, “Measuring topological invariants from generalized edge states in polaritonic quasicrystals,” Phys. Rev. B 95(16), 161114 (2017).
[Crossref]

Gao, W.

C. Liu, W. Gao, B. Yang, and S. Zhang, “Disorder-induced topological state transition in photonic metamaterials,” Phys. Rev. Lett. 119(18), 183901 (2017).
[Crossref] [PubMed]

Gao, W. S.

Genov, D. A.

S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101(4), 047401 (2008).
[Crossref] [PubMed]

Gneiding, N.

E. Tatarschuk, N. Gneiding, F. Hesmer, A. Radkovskaya, and E. Shamonina, “Mapping inter-element coupling in metamaterials: Scaling down to infrared,” J. Appl. Phys. 111(9), 094904 (2012).
[Crossref]

Gómez, C.

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L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological states in photonic systems,” Nat. Phys. 12(7), 626–629 (2016).
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M. Xiao, G. Ma, Z. Yang, P. Sheng, Z. Q. Zhang, and C. T. Chan, “Geometric phase and band inversion in periodic acoustic systems,” Nat. Phys. 11(3), 240–244 (2015).
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A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2013).
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S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, “Topologically protected bound states in photonic parity-time-symmetric crystals,” Nat. Mater. 16(4), 433–438 (2017).
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M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photonics 7(12), 1001–1005 (2013).
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M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photonics 7(12), 1001–1005 (2013).
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A. Kurs, A. Karalis, R. Moffatt, J. D. Joannopoulos, P. Fisher, and M. Soljačić, “Wireless power transfer via strongly coupled magnetic resonances,” Science 317(5834), 83–86 (2007).
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M. Bellec, U. Kuhl, G. Montambaux, and F. Mortessagne, “Tight-binding couplings in microwave artificial graphene,” Phys. Rev. B 88(11), 115437 (2013).
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C. Poli, M. Bellec, U. Kuhl, F. Mortessagne, and H. Schomerus, “Selective enhancement of topologically induced interface states in a dielectric resonator chain,” Nat. Commun. 6(1), 6710 (2015).
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A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2013).
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S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, “Topologically protected bound states in photonic parity-time-symmetric crystals,” Nat. Mater. 16(4), 433–438 (2017).
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C. Poli, M. Bellec, U. Kuhl, F. Mortessagne, and H. Schomerus, “Selective enhancement of topologically induced interface states in a dielectric resonator chain,” Nat. Commun. 6(1), 6710 (2015).
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E. Tatarschuk, N. Gneiding, F. Hesmer, A. Radkovskaya, and E. Shamonina, “Mapping inter-element coupling in metamaterials: Scaling down to infrared,” J. Appl. Phys. 111(9), 094904 (2012).
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F. D. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100(1), 013904 (2008).
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Z. Guo, H. Jiang, Y. Long, K. Yu, J. Ren, C. Xue, and H. Chen, “Photonic spin Hall efect in waveguides composed of two types of single-negative metamaterials,” Sci. Rep. 7(1), 7742 (2017).
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F. Baboux, E. Levy, A. Lemaître, C. Gómez, E. Galopin, L. L. Gratiet, I. Sagnes, A. Amo, J. Bloch, and E. Akkermans, “Measuring topological invariants from generalized edge states in polaritonic quasicrystals,” Phys. Rev. B 95(16), 161114 (2017).
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C. Poli, M. Bellec, U. Kuhl, F. Mortessagne, and H. Schomerus, “Selective enhancement of topologically induced interface states in a dielectric resonator chain,” Nat. Commun. 6(1), 6710 (2015).
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S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, “Topologically protected bound states in photonic parity-time-symmetric crystals,” Nat. Mater. 16(4), 433–438 (2017).
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A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2013).
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A. P. Slobozhanyuk, A. N. Poddubny, A. E. Miroshnichenko, P. A. Belov, and Y. S. Kivshar, “Subwavelength topological edge States in optically resonant dielectric structures,” Phys. Rev. Lett. 114(12), 123901 (2015).
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L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological states in photonic systems,” Nat. Phys. 12(7), 626–629 (2016).
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Y. Hadad, J. C. Soric, A. B. Khanikaev, and A. Alu, “Self-induced topological protection in nonlinear circuit arrays,” Nat. Electron. 1(3), 178–182 (2018).
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S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, “Topologically protected bound states in photonic parity-time-symmetric crystals,” Nat. Mater. 16(4), 433–438 (2017).
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Tan, W.

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M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photonics 7(12), 1001–1005 (2013).
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A. B. Khanikaev, S. H. Mousavi, W. K. Tse, M. Kargarian, A. H. MacDonald, and G. Shvets, “Photonic topological insulators,” Nat. Mater. 12(3), 233–239 (2013).
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S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A. Szameit, “Topologically protected bound states in photonic parity-time-symmetric crystals,” Nat. Mater. 16(4), 433–438 (2017).
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Z. Guo, H. Jiang, Y. Long, K. Yu, J. Ren, C. Xue, and H. Chen, “Photonic spin Hall efect in waveguides composed of two types of single-negative metamaterials,” Sci. Rep. 7(1), 7742 (2017).
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Figures (8)

Fig. 1
Fig. 1 Experimental setup. The one dimensional dimer chain composed by equally spaced 32 identical SRRs (not shown all the SRRs) is arranged on a foam substrate, and sandwiched by two metallic plates in experiments (here the top metal plate is taken away in order to take a picture of the chain). The near-field probe made of a non-resonant loop is used to measure the density of states.
Fig. 2
Fig. 2 (a) Sketch of a pair of arbitrarily rotated split rings. (b) Controlling coupling strengthen realized by tuning the relevant angle between two resonators. The strong and weak coupling strength used in our SSH chains are shown in the upper inset and down inset respectively.
Fig. 3
Fig. 3 Two types of dimer chains differ in their topological properties. (a-b) Schematic of the unit cell, calculated Eigen frequencies (black dots), and measured DOS spectrum (blue profile) of the Type I and the Type II chains are given in (a) and (b), respectively. (c) Experimental (red triangles) and theoretical (grey dashed line) LDOS distribution of the edge state (f = 1.9GHz). (d) LDOS profiles of the ordinary state in pass band (f = 1.95GHz).
Fig. 4
Fig. 4 Robust edge state against the losses perturbation. (a) The topological nontrivial chain with losses added into the central 20 SRRs (indicated by the grey background). (b) Measured DOS spectrum with loss. DOS of the edge state is much more robust than that of the bulk state. (c) Measured LDOS distribution of the edge state (f = 1.9GHz) with losses (red triangles), along with the theoretical calculations without losses (grey dashed line). (d) Measured LDOS distribution of the ordinary bulk state (f = 1.95GHz) with losses (red triangles), along with the theoretical calculations without losses (grey dashed line).
Fig. 5
Fig. 5 Robust edge states against the disorder perturbation. (a) Schematic representation of the topological nontrivial chain with disorder (random coupling strengths by rotating the central 20 SRRs). Detail of rotation is shown in the inset of Fig. 4(b). (b, c) Measured LDOS distributions of the edge state and the bulk state at various disorder levels, including α = 1 ° (red circles), α = 3 ° (blue stars), and α = 5 ° (green triangles). As a comparison, the calculated LDOS distribution of the edge (f = 1.9GHz) and the bulk (f = 1.95GHz) states in the original chain are also presented (gray dashed line).
Fig. 6
Fig. 6 Measured LDOS distributions of the edge states in the chain with both losses and disorder perturbations in the central 20 SRRs. As a comparison, the calculated LDOS distribution of the edge state (f = 1.9GHz) in the original chain are also presented (grey dashed line).
Fig. 7
Fig. 7 (a) The schematic of a single SRR. (b) A pair of coupled SRRs in the planar configuration. The angle of rotation of two resonators are marked by φ 1 and φ 2 .
Fig. 8
Fig. 8 The relationship between κ s l (l = 1, 2, 3, 4, and 5) and the rotation angle of the pair of SRRs.

Equations (21)

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ω n o r 2 ( a k b k ) = ( 1 κ intra + κ inter e i k d κ intra + κ inter e i k d 1 ) ( a k b k ) ,
w = 1 2 π π / d π / d θ k d k ,
M = μ 0 4 π I m 1 ( r 1 ) I m 2 ( r 2 ) | r 1 r 2 | d s 1 d s 2 ,
K 1 = 1 4 π ε 0 ρ m 1 ( r 1 ) ρ m 2 ( r 2 ) | r 1 r 2 | d τ 1 d τ 2 ,
ρ m ( θ ) = ( tan θ 2 ) / 2 ln ( g 4 r ) ,
θ < θ g
I m ( θ ) = ln ( cos θ g 2 cos θ 2 ) ln 1 ( cos θ g 2 ) .
L = μ 0 r ( ln 8 r h + w 1 2 ) ,
C = ε 0 [ ( w + g ) ( h + g ) g + 2 ( h + w ) π ln ( 4 r g ) ] .
κ H = 2 M L , κ E = 2 C K .
l = n [ L 2 ( p ˙ n 2 + q ˙ n 2 ) p n 2 + q n 2 2 C + M 2 p ˙ n q ˙ n 1 p n q n 1 K 2 p n q n K 1 + M 1 p ˙ n q ˙ n ] .
d d t ( l α ˙ n ) l α n = 0 , ( α = p , q ) .
d d t ( L p ˙ n + M 2 q ˙ n 1 + M 1 q ˙ n ) + 1 i ω C p ˙ n + 1 i ω K 2 q ˙ n 1 + 1 i ω K 1 q ˙ n = 0 d d t ( L q ˙ n + M 2 p ˙ n + 1 + M 1 p ˙ n ) + 1 i ω C q ˙ n + 1 i ω K 2 p ˙ n + 1 + 1 i ω K 1 p ˙ n = 0 .
ω n o r 2 [ κ H 2 2 + κ H 1 2 + 2 cos ( k d ) κ H 2 κ H 1 1 ] ( a k b k ) = H ' ( a k b k ) .
ω n o r 2 ( a k b k ) = ( 1 κ 1 + κ 2 e i k d κ 1 + κ 2 e i k d 1 ) ( a k b k ) ,
ω n o r 2 ( a k b k ) = D k ( a k b k ) ,
| u k , ± = ( a k b k ) = ( κ 1 + κ 2 cos ( k d ) i κ 2 sin ( k d ) ( κ 1 + κ 2 cos ( k d ) ) 2 + ( κ 2 sin ( k d ) ) 2 1 ) ,
| u k , ± = ( e i θ k 1 ) ,
| u k , ± = 1 2 ( e i θ k 1 ) ,
w w i n d i n g = i π π / d π / d ( a k * k a k + b k * k b k ) d k .
w = i π π / d π / d ( a k * k a k + b k * k b k ) d k = 1 2 π π / d π / d θ k d k ,

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